The proof uses standard results on convex multi-objective optimization from [1]}, [2]}. Written in the form (REF ), the GGFs corresponds to the scalarization of the multi-objective problem of jointly maximizing the generalized Lorenz curves of users and items, which is a problem with \(n+m\)  objectives. Indeed, each objective function is a point of the generalized Lorenz curve \(({U}(P),{V}(P)).\)  Each objective \(U_i(P)\)  is concave because it corresponds to an OWA operator with non-increasing weights \({\rho }\)  with \(\rho _{i^{\prime }} = {i^{\prime }\le i},\)  applied to utilities, which are linear functions of the ranking policy. Each objective \(V_i(P)\)  is similarly concave. Moreover, we are optimizing over the convex set of stochastic ranking policies \(\mathcal {P}\) . The multi-objective problem is then concave, which means that the maximizers of all weighted sums of the objectives \(({U}(P),{V}(P))\)  with strictly positive weights are Pareto-efficient. Reciprocally every Pareto-efficient solution is a solution of a non-negative weighted sum of the objectives \(({U}(P),{V}(P)),\)  where the weights sum to 1 [2]}.
